Index theorems and superconducting cosmic strings

Abstract

The interaction of vortex lines with chiral fermions in 3+1 spacetime dimensions is investigated. We construct Dirac operators relevant to the study of superconducting cosmic strings, obtain conditions for the cancellation of triangle anomalies, and examine properties of the corresponding Dirac Hamiltonian. Our analysis is applicable to both cosmic strings arising from spontaneously broken gauged U(1) symmetries and axion vortices in broken global U(1) symmetry groups. We generalize known index theorems to consider angular-momentum-weighted indices and η invariants of the Hamiltonian in the background field of a topological vortex. We further obtain explicit zero modes of the Hamiltonian in a rotationally covariant vortex field. Implications of the results for the quantum numbers of light fermionic excitations and their axial anomalies are discussed. We use the η invariants to derive anomaly equations for charges and angular momenta and find discrepancies with those of effective two-dimensional field theory. Our results indicate a novel anomalous behavior of the angular momentum and suggest a new mechanism for the transfer of energy and momentum between axionic strings.

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